\section{Theoretical Implementation}
\paragraph{}  A peer grading system would allow for more efficient and 
potentially more constructive feedback, wherein students get the added 
benefit of reviewing each others work.  Making it anonymous would 
prevent the students from taking advantage of the system, and a 
reputation scheme provides incentive to grade accurately.  Students' 
final assignment grades are calculated based on the scores given by 
graders, weighted by the graders' reputation scores.  A student's own 
reputation score does not affect his or her final assignment grade, so 
to maintain incentive to grade accurately, a student reputation score 
must be a significant portion of their overall grade for the class.

\paragraph{}When students submit an assignment, they are required to 
give themselves a grade.  The system then randomly distributes submitted 
assignments to TAs and other students.  The professor would also be 
randomly assigned a reasonable portion of the assignments to grade.  
Since the reputation of the professor is static at the maximum possible 
reputation score, the grades that the professor gives would count for 
the paper's entire grade. However, for the papers that the professor did 
not grade, the grade would be based on the peer and teaching assistant 
grades.  More specifically, each grader's reputation is multiplied by 
the grade they gave the assignment and multiplied by the inverse of the 
total reputation score (the sum of all graders' reputations for each 
assignment.)  The overall grade is the sum of
these products.  For example, given two peer grades and one TA grade, if 
their reputations are all equal then their grades would be equally 
weighted in the average to determine the final grade.  But, if the 
teaching assistant has a rating of 6 and each peer has a rating of 3, 
the teaching assistant's grade will count twice as much as each peer's.  
The exact formula for this calculation is written below, where $R$ 
represents the reputation score and $g$ represents numerical grades.

\begin{center}
$g_{overall} = \dfrac{1}{R_{total}}\displaystyle\sum_{i=0}^n R_i g_i$
\end{center}

\paragraph{}The reputation scheme is much less trivial.  To start, we 
used a scale from zero to ten, where ten is the most reputable grader.  
The professor would be fixed at ten, as her or his opinion is final.  
The students would start with a rating of one, and the teaching 
assistants would begin with a rating of seven.  Reputation, $R$, 
increases or decreases based on how much each grader (including 
self-grading) matches the grade given by the grader with the highest 
reputation, as shown below, where $\Delta_g$ is the absolute value of 
the difference between the numerical grade (1 to 100) of the most 
reputable grader and each individual grader and $\Delta_r$ is the 
absolute value of the difference between the reputation of the two 
graders.  If the difference is less than or equal to seven, the graders 
of lower reputation gain reputation points.  For numerical grades more 
than seven points apart, reputation is lost.  

\begin{displaymath}
\begin{aligned}
   f(\Delta_g) &= \left\{
     \begin{array}{lr}
       -\frac{3}{70} \Delta_g + \frac{1}{2} & : \Delta_g \le 7\\
       -\frac{1}{66}(\Delta_g - 7) & : \Delta_g > 7
     \end{array}
   \right. \\
   R_n &= R_{n-1} + \Delta_r f(\Delta_g)
\end{aligned}
\end{displaymath} 


\paragraph{}  This equation implies that the rate of reputation increase 
is greater
than the rate of reputation loss.  However, the target range for gaining 
reputation (where $\Delta_g \le 7$ )
is a narrow target area.  The equation also implies that at the maximum 
$\Delta_g$
(which is 40 and can only occur when a professor gives a 100 and a 
student or TA gives a 60 or vice versa)
the grader will lose at most $\frac{1}{3}$ of his or her reputation.
One will note that there is no increase to the reputation of the highest 
rated grader.  It was decided to not give them additional reputation 
points because they were being used as the basis for every other rating 
increase.  Additionally, one will eventually grade a paper that has also 
been assigned to the professor, so even the elite TAs and student 
graders will have opportunities to further increase their reputation.  
It is, however, impossible to reach the perfect reputation rating of the 
instructor.  Given the scenario where a student repeatedly matches the 
grade of the instructor exactly, a plot of their reputation over time 
would approach 10 at infinity.  In practical terms, this system allows 
students to gain reputation quickly early on, but it is difficult to 
gain more reputation as one's rating rises.  

\begin{figure}[htp]
\centering
\includegraphics[width=\textwidth]{reputationFig1}
\caption{Changes to Reputation Score Depending on Grade 
Differences}%\label{fig:}
\end{figure}

\paragraph{}As shown in the figure and equations above, the reputation 
increases at a rate
much greater than it decreases, but range for matching is rather low, percentage wise. 
In more ordinary terms, the center of the target is small, but the reward for hitting
it is much greater than the consequences for missing.  Thus, if one is honestly grading,
overtime their rank is very likely to increase over time, even if their opinions differ
from the professor or teaching assistants.  This will also encourage more objective grading
by all graders.  

\paragraph{}One current question in this model is whether or not grades should be retroactively
changed based on changes in the reputations of graders.  In our system, final grades are calculated once
and do not change even though graders' reputations may change.  We conclude that students would perceive that
the system was unfair if their grade continuously changed (even if slightly).  It would also require more processing time to update
the entire database of grades each time a grader's reputation changed.  However, a retroactive system may be a more accurate depiction of grading quality since
the assignment sample size would be larger and the reputation of a grader over time would be more accurate than a reputation score
which has been generated by only a few assignments.  


\paragraph{}Taking a step back, these methods will undoubtedly require 
some fine tuning based on the number of assignments, ratio of students 
to TAs, and raw number of students.  Furthermore, this system is 
certainly not infallible, but measures have been taken to account for 
many scenarios of academic dishonesty.  Students would not be able to 
just agree to give each other high grades, since they would risk a 
penalty to their reputation score if they grade the same assignment that 
is graded by a ``trusted'' grader such as a professor or TA.
One important feature that cannot be eschewed however, is that students 
must be accountable for their reputation grades at the end of the 
semester.
Since anonymous grading systems are uncommon, it is necessary to test 
these theories against student participation, performance, and reactions 
to determine its effectiveness.
